Search Results (131)

Let $\mathcal{A}=\{p+Jq:p,q\in \mathbb{C},J^2=-1\}$ be the commutative bicomplex-type algebra in which J commutes with the scalar imaginary unit. A Cauchy–Riemann-type operator $\overline{D}$ is studied on domains in ${\mathbb{C}}^2$. In the active coordinates $\xi =z_1-i{z}_2$ and $\eta =z_1+i{z}_2$, the equation $\overline{D}f=0$ is diagonal in the idempotent basis: the $e_{+}$-component is holomorphic in $\xi$ with $\eta$ as the parameter, while the $e_{-}$-component is holomorphic in $\eta$ with $\xi$ as the parameter. The expression $e_{+}F\left(\xi \right)+e_{-}G\left(\eta \right)$ is the parameter-independent subcase. From this decomposition, one obtains a slice characterization, a criterion for separatedness, a comparison with ordinary holomorphic functions of two complex variables, active-variable Cauchy formulas and estimates, local series with parameter-dependent coefficients, reflection symmetry, and Hardy and Bergman kernel lifts on the separated Hilbert spaces. Full article

The proliferation of AI-generated content and personalized AI systems has sharpened two fundamental and related computational problems: the progressive erosion of authentic identity in AI-mediated representations, and the growing difficulty of distinguishing human-originated from AI-generated behavioral and textual streams. This paper proposes a rigorous computational framework in which digital identity is formalized as a holomorphic function of complex time T = (a + ib) ∈ ℂ, where the real component Re$\left(T\right)$ encodes chronological progression and the imaginary component Im$\left(T\right)$ spans a continuum from episodic memory ($Im\left(T\right) < 0$) through the present moment ($Im\left(T\right) = 0$) to prospective imagination ($Im\left(T\right) > 0$). We argue that holomorphicity—enforced via Cauchy–Riemann regularization during CTNN learning (Proposition 1)—provides a theoretically grounded encoding of identity coherence, and discuss its advantages over alternative mathematical choices, including Lipschitz continuity, ${\mathit{C}}^{\mathbf{\infty }}$ smoothness, piecewise analytic functions, and stochastic models. Under four explicit Assumptions 1–4 covering the Markovian structure and fixed context window of current LLM architectures, we establish via Lemmas 1 and 2 and Theorem 1 that AI-generated behavioral trajectories exhibit structural limitations in satisfying the Cauchy–Riemann conditions at temporal depths characteristic of human biographical memory—limitations that do not arise for human trajectories learned under CTNN regularization. Building on this result, we introduce the Human–AI Authenticity Discriminant (HAAD), a theoretically grounded classifier with a fully specified calibration algorithm and sensitivity analysis ($\mathit{\kappa }$ ΔAUROC ≤ 0.04 over ±30% perturbation). Five metrics—TCS, ISI, PAS, GAS, and HAAD—are derived analytically from the holomorphic structure. The algorithmic framework is instantiated on four real-world datasets: MovieLens 25M, the Pushshift Reddit corpus, the Stack Overflow Data Dump, and the LIAR dataset. On the LIAR benchmark, TDT-HAAD achieves AUROC = 0.82 (95% CI: [0.79, 0.85]), exceeding a RoBERTa-based LLM detector baseline (AUROC = 0.75, DeLong p < 0.01); an ablation study supports the structural contribution of each component. A credibility harvesting signature is detectable 45.3 ± 12.1 days before standard temporal models reach statistical significance. Full article

This paper extends Loomis’s classical spectral criterion for almost periodic functions to bounded sequences in discrete settings. We first establish a discrete Kadets-type theorem for bounded sequences by regarding almost periodic functions as continuous with respect to the Bohr metric. We then introduce three spectra for bounded sequences via the Carleman transform and prove their equivalence. Using the Beurling–Gelfand theorem, we derive a discrete Loomis-type theorem, providing a spectral criterion for almost periodicity of bounded sequences. Our results extend the continuous theory to the discrete case and offer new tools for analyzing almost periodicity in sequence spaces. Full article

This paper develops a structured analytical framework and a robust numerical methodology for the spectral stability of time-varying bilateral quaternion differential equations of the form $\dot{q}=A\left(t\right)q+qB\left(t\right)$. By systematically extending classical real matrix theory to non-commutative dynamical systems via exact isometric real representations, this study utilizes the Kronecker product of real adjoint matrices to rigorously elucidate the underlying tensor structure of the bilateral evolution operator. This tensor-based reformulation proves that the Floquet multipliers of the bilaterally coupled system can be strictly decoupled into the product of the spectra corresponding to the left and right unilateral subsystems. Second, a “Scalar-Vector Stability Separation Principle” based on logarithmic norms is proposed, demonstrating that the transient energy evolution of the system is governed exclusively by the Hermitian real parts of the coefficient matrices, remaining entirely independent of the anti-Hermitian imaginary parts (rotation terms). Furthermore, for constant-coefficient and slowly varying systems, the Riesz projection from holomorphic functional calculus is introduced to establish algebraic criteria for exponential dichotomies, thereby revealing a cubic scaling law that relates the robustness threshold to the spectral gap (${\epsilon }_0\propto {\beta }^3$). Numerically, a Quaternion Chebyshev Spectral Collocation Method (Q-CSCM) is embedded within this exact vectorization framework to ensure that the algebraic symmetries of the bilateral system are strictly preserved through the isomorphic mapping. By explicitly constructing the fully discrete Kronecker product matrix via the exact real vectorization isomorphism, discrete energy estimates are utilized to rigorously prove that the numerical scheme successfully inherits the intrinsic spectral accuracy of the Chebyshev approximation. Comprehensive numerical experiments demonstrate that, within the low-dimensional regime, this methodology exhibits substantial temporal approximation efficiency advantages and superior numerical robustness compared to an alternative Legendre spectral baseline, as well as traditional explicit and state-of-the-art implicit symplectic Runge–Kutta methods, particularly when solving stiff and critically stable problems such as nonlinear Riccati oscillators. Full article

In this paper, we study a generalization of the Graham–Kohr extension operator, ${\mathsf{\Psi }}_{n,\alpha ,\beta }^{\gamma }\left(f\right)$, which maps functions defined on the unit disk into holomorphic mappings in the unit ball ${\mathbb{B}}^n$. Using the theory of Loewner chains, we show that, under suitable conditions, this operator can be embedded as the first element of a Loewner chain while preserving geometric properties. In addition, for suitable choices of the parameters, we establish subordination relations among starlike functions. Full article

Motivated by the fact that the Fibonacci sequence is the simplest nontrivial second-order recurrence with a rational generating function, we develop a Fibonacci-weighted Hardy theory for bicomplex holomorphic functions. Starting from the coefficient norm ${\sum }_{n\ge 0}{\left|a_n\right|}^2/F_{n+1}$, we obtain a bicomplex Hilbert module whose reproducing kernel is governed by ${(1-t-t^2)}^{-1}$ and whose maximal disk of holomorphy is determined sharply by the nearest kernel singularity, giving the radius ${\rho }_F={\phi }^{-1/2}$ (the square-root inverse of the golden ratio $\phi$). The arithmetic recurrence makes several objects fully explicit: we derive closed formulas for the kernels through the idempotent decomposition of $\mathbb{BC}$, compute exact norms of the shift powers and a golden-ratio spectral radius, and package the local theory into a sheaf of Fibonacci-holomorphic germs that are compatible with the bicomplex idempotent splitting. We also treat $(p,q)$-Fibonacci weights, obtaining a one-parameter family of rational kernels ${(1-pt-q{t}^2)}^{-1}$ and corresponding operator bounds. In addition to providing a concrete bicomplex model within weighted Hardy theory, the resulting explicit kernels furnish benchmark examples for kernel-based interpolation and for the operator theory of unilateral weighted shifts. Full article

In this paper, we introduce a new logarithmic integral operator that unifies differentiation and fractional integration within the complex domain. The present work addresses this gap by applying the proposed operator to analytic functions represented by alternating power series. The method demonstrates that the coefficients can be reorganized in a controlled manner without affecting convergence or analytic behavior. Using this framework, we derive third-order differential subordination and superordination results, which naturally lead to corresponding sandwich-type results. The findings confirm that the introduced operator offers an effective analytical tool for studying distortion, growth, and mapping properties of analytic functions, with promising potential for future applications in fluid mechanics. Full article

Using the instanton partition function for five-dimensional $U\left(1\right)$ gauge theory with eight supercharges and a single adjoint massive hypermultiplet on the $\Omega$ background, we give explicit expression for non-perturbative corrections to the topological string theory in the holomorphic limit. It was argued that in this case the theory is compactified on the twisted affine line bundle over $\mathbb{C}\times T^2$. We perform calculations in two ways. First we modify the integration contour by adding poles responsible for non-perturbative physics in accordance with a recent proposal. Then, we compute the genus zero Gopakumar–Vafa invariants for our case and evaluate the non-perturbative corrections to the partition function. We check that both calculations give the same result. Full article

The q-calculus framework has emerged as a powerful tool in geometric function theory, enabling refined analysis of analytic and bi-univalent functions. Inspired by the versatility of the q-derivative operator, this paper introduces a new generalized subclass of bi-univalent functions defined via the q-derivative in combination with generalized bivariate Fibonacci polynomials, which have recently gained significant attention in mathematical research. For functions in this class, we establish bounds on the initial coefficients and provide estimates for the corresponding Fekete–Szegö functional. By appropriate specialization of parameters, our results recover several known findings and, importantly, produce bounds for new subclasses of bi-univalent functions not previously studied. This framework unifies earlier developments while extending the theory to novel, analytically meaningful classes. Full article

This work introduces a new initialization scheme for complex-valued layers in physics-informed neural networks that use holomorphic activation functions. The proposed method is derived empirically by estimating the activation and gradient gains specific to complex-valued tanh and sigmoid functions through Monte Carlo simulations. These estimates are then used to formulate variance-preserving initialization rules. The effectiveness of these formulas is evaluated on several second-order complex-valued ordinary differential equations derived from the Helmholtz equation, a fundamental model in wave theory and theoretical physics. Comparative experiments show that complex-valued neural solvers initialized with the proposed method outperform traditional real-valued physics-informed neural networks in terms of both accuracy and training dynamics. Full article

This paper broadens the scope of existing research: the shared value is generalized from a non-zero finite complex number to a non-identically zero holomorphic function, the order of the derivative is extended from the first order to an arbitrary k-th order, and the constraint condition on the polynomial H is simplified to $degH\ge 2$. A more general normality criterion for families of meromorphic functions involving the sharing of differential polynomials is proved. Let D be a domain, $\mathcal{F}$ be a family of meromorphic functions in D, and $P\left(z\right)$ be a non-identically zero holomorphic function in D. If for any $f,g\in \mathcal{F}$, the differential polynomials $H\left(f\right)f^{\left(k\right)}$ and $H\left(g\right)g^{\left(k\right)}$ share $P\left(z\right)$ in D, then $\mathcal{F}$ is normal in D. Full article

This paper explains why the critical line sits at the real part equal to one-half by treating it as an intrinsic boundary of a reparametrized complex plane (“$z$-space”), not a mere artifact of functional symmetry. In $z$-space the real part is defined by a geometric-series map that gives rise to a rulebook for admissible analytic operations. Within this setting we rederive the classical toolkit—the eta–zeta relation, Gamma reflection and duplication, theta–Mellin identity, functional equation, and the completed zeta—without importing analytic continuation from the usual $s$-variable. We show that access to the left half-plane occurs entirely through formulas written on the right, with boundary matching only along the line with the real part equal to one-half. A global Hadamard product confirms the consistency and fixed location of this boundary, and a holomorphic change of variables transports these conclusions into the classical setting. Full article

We investigated an exact solution in a conformal invariant Randall-Sundrum 5D warped brane world model on a time dependent Kerr-like spacetime. The singular points are determined by a quintic polynomial in the complex plane and fulfills Cauchy’s theorem on holomorphic functions. The solution, which is determined by a first-degree differential equation, shows many similarities with an instanton. In order to describe the quantum mechanical aspects of the black hole solution, we apply the antipodal boundary condition. The solution is invariant under time reversal and also valid in Riemannian space. Moreover, CPT invariance in maintained. The vacuum instanton solution follows from the 5D as well as the effective 4D brane equations, only when we allow the contribution of the projected 5D Weyl tensor on the brane (the KK-‘particles’). The topology of the effective 4D space of the brane is the projective $\mathbb{R}{P}^3$ (elliptic space) by identifying antipodal points on $S^3$. The 5D is completed by applying the Klein bottle embedding and the ${\mathbb{Z}}_2$ symmetry of the RS model. This model fits very well with the description of the Hawking radiation, which remains pure. We have also indicated a possible way to include fermions. Our 5D space admits a double cover of $S^3$ and after fibering to the $S^2$, we obtain the effective black hole horizon. The connection with the icosahedron discrete symmetry group is investigated. It seem that Bekenstein’s conjecture that the area of a black hole is quantized, could be applied to our model. Full article

The first aim of this study is to point out new aspects of approximation theory applied to a few classes of holomorphic functions via Vitali’s theorem. The approximation is made with the aid of the complex moments of the functions involved, which are defined similarly to the moments of a real-valued continuous function. By applying uniform approximation of continuous functions on compact intervals via Korovkin’s theorem, the hard part concerning uniform approximation on compact subsets of the complex plane follows according to Vitali’s theorem. The theorem on the set of zeros of a holomorphic function is also applied. In the end, the existence and uniqueness of the solution for a multidimensional moment problem are characterized in terms of limits of sums of quadratic expressions. This is the application appearing at the end of the title. Consequences resulting from the first part of the paper are pointed out with the aid of functional calculus for self-adjoint operators. Full article